Incompleteness and the Barcan formula

Journal of Philosophical Logic 24 (4):379 - 403 (1995)
A (normal) system of propositional modal logic is said to be complete iff it is characterized by a class of (Kripke) frames. When we move to modal predicate logic the question of completeness can again be raised. It is not hard to prove that if a predicate modal logic is complete then it is characterized by the class of all frames for the propositional logic on which it is based. Nor is it hard to prove that if a propositional modal logic is incomplete then so is the predicate logic based on it. But the interesting question is whether a complete propositional modal logic can have an incomplete extension. In 1967 Kripke announced the incompleteness of a predicate extension of S4. The purpose of the present article is to present several such systems. In the first group it is the systems with the Barcan Formula which are incomplete, while those without are complete. In the second group it is those without the Barcan formula which are incomplete, while those with the Barcan Formula are complete. But all these are based on propositional systems which are characterized by frames satisfying in each case a single first-order sentence
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DOI 10.1007/BF01048353
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References found in this work BETA
Semantical Considerations on Modal Logic.Saul A. Kripke - 1963 - Acta Philosophica Fennica 16 (1963):83-94.
Incompleteness Results in Kripke Semantics.Silvio Ghilardi - 1991 - Journal of Symbolic Logic 56 (2):517-538.

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Partly Free Semantics for Some Anderson-Like Ontological Proofs.Mirosław Szatkowski - 2011 - Journal of Logic, Language and Information 20 (4):475-512.

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